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98 Chapter 3 ■ Digital Morphology

The proof of the erosion-dilation duality is fairly simple and may yield some
insights into how morphological expressions are manipulated and validated.
The definition of erosion is:

A B ={z|(B) z ⊆ A} (EQ 3.13)
So, the complement of the erosion is:

c c
(A B) ={z|(B) z ⊆ A} (EQ 3.14)
If (B) z is a subset of A, then the intersection of (B) z with A is not empty:

c c
(A B) ={z|((B) z ∩ A) = ∅} (EQ 3.15)
c
but the intersection with A will be empty:
c
c
z|(B) z ∩ A = ∅ (EQ 3.16)
and the set of pixels not having this property is the complement of the set
that does:

c
z|((B) z ∩ A ) = ∅ (EQ 3.17)
c
By the definition of translation in Equation 3.1, if (B) z intersects A ,then
c
{z|b + z ∈ A , b ∈ B} (EQ 3.18)
which is the same thing as
c
{z|b + z = a, a ∈ A , b ∈ B} (EQ 3.19)
Now if a = b + z,then z = a − b:
c
{z|z = a − b, a ∈ A , b ∈ B} (EQ 3.20)
Finally, using the definition of reflection, if b is a member of B,then −b is a
member of the reflection of B:
c ˆ
{z|z = a + b, a ∈ A , b ∈ B} (EQ 3.21)
ˆ
c
which is the definition of A ⊕ B.
The erosion operation also brings up an issue that was not a concern about
dilation: the idea of a ‘‘don’t care’’ state in the structuring element. When using
a strictly binary structuring element to perform an erosion, the member black
pixels must correspond to black pixels in the image in order to set the pixel
in the result, but the same is not true for a white (0) pixel in the structuring
element. We don’t care what the corresponding pixel in the image might be
when the structuring element pixel is white.
Figure 3.8 gives some examples of erosions of a simple image by a collection
of different structuring elements. The basic shape of the structuring element